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For $n \ge 1$, let $$a_n = \dfrac1{1 \cdot 5} + \dfrac1{5 \cdot 9} + \cdots + \dfrac1{(4n-3)(4n+1)}.$$ Guess a simple explicit formula for $a_n$ and prove it by induction.

Hi, I'm trying to answer this question. I was not provided with a solution.

Kenny Lau
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studymaths
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    Have you tried computing it for a few values of $n$ and tried to guess the pattern? – Angina Seng May 15 '17 at 05:51
  • Im still keep trying but something is off everytime. i dont have much mathematical background so im really struggling with these kind of proofs. i was hoping to get a solution for this so i can see how i can tackle this problem. thanks – studymaths May 15 '17 at 05:54
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    @arcadeboy Compute $a_1$, $a_2$, and $a_3$. Edit them into your question. – Kenny Lau May 15 '17 at 05:54
  • Can someone show me how to solve this? – studymaths May 15 '17 at 05:58

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$$a_n=\frac{1}{4}\sum_{k=1}^n\left(\frac{1}{4k-3}-\frac{1}{4k+1}\right)=$$ $$=\frac{1}{4}\left(1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+...+\frac{1}{4n-3}-\frac{1}{4n+1}\right)=$$ $$=\frac{1}{4}\left(1-\frac{1}{4n+1}\right)=\frac{n}{4n+1}$$

Michael Rozenberg
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